By \(\cite {punc}\) , to a tropical type \(\tau \) , there is an associated DM log stack \(\mathcal {M}(X,\tau )\)  marked by \(\tau \)  in the sense of Definition \(3.7\)  of [1]}. When \(\tau \)  is realizable, this condition on a geometric point \((C^\circ ,f)\)  is equivalent to the condition that the tropical type of \((C^\circ ,f)\)  has an associated universal tropical moduli space \(\tau ^{\prime }\)  which contains a face \(\tau \subset \tau ^{\prime }\) , such that the restriction of the universal family \(\Gamma _{\tau ^{\prime }} \rightarrow \tau ^{\prime }\)  to the face \(\tau \)  is the universal family of the realizable tropical type \(\tau \) . The moduli stack \(\mathcal {M}(X,\tau )\)  in general possesses the typical singular behavior that is standard in a moduli stack of maps, and in order to construct a useful perfect obstruction theory which allows for the definition of log Gromov Witten invariants, we make further use of the tropical theory.
